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CHAPTER 6
ANALYSIS OF MATRIX CONVERTERS
6.1 INTRODUCTION
New technical challenges emerge due to increased wind power
penetration, dynamic stability and power quality, implying research of more
realistic physical models for wind energy systems. Power electronic interfaces
have been developed for integrating wind turbine generator with the grid or to
the load. The use of power electronic converters in variable speed wind
energy conversion scheme yields enhanced power extraction. In variable
speed operation, a control method designed to extract maximum power from
the turbine and provide constant grid voltage and frequency is required
(Baroudi et al 2007).
Recently there has been considerable interest in the potential
benefits of matrix converter technology, especially for variable speed
induction machine applications (Ghedamsi et al 2006 and Wheeler et al
2002). The matrix converter is an advanced circuit topology capable of
converting AC to AC. As stated by Venturini (1980), many benefits are
encompassed by this topology such as, the matrix converter fulfills the
requirements to provide a sinusoidal voltage at the load side and on the other
hand, it is also possible to adjust the unity power factor on the mains side
under certain conditions. Since there is no DC link as in common converters,
the matrix converter can be built as a full-silicon structure. Using a
134
sufficiently high pulse frequency, the output voltage and input current both
are shaped sinusoidally.
The matrix converter is an alternative to an inverter drive for three-
phase frequency control. This chapter deals with the implementation and
analysis of single phase and three phase matrix converters employing
different control algorithms applicable for variable speed wind energy
conversion systems supplying isolated loads.
6.2 SINGLE PHASE MATRIX CONVERTER
Single phase AC–AC converter has a number of potential
applications. Several approaches to control the output of single-phase AC-AC
converter are available in the literature (Jacobina et al 2006, Perez et al 2006,
Sunter and Aydogmus 2008). The proposed work deals with a design,
simulation and implementation of single-phase induction motor drive fed by a
Sinusoidal PWM (SPWM) based matrix converter used in variable speed
applications. The characteristics of a single phase AC-AC converter feeding
an induction motor load with particular emphasis on the harmonic content is
presented with the aid of simulation study and hardware implementation.
6.1.1 Converter Topology
Figure 6.1 shows the block diagram of single phase matrix
converter fed induction motor drive. The matrix converter converts fixed
frequency fixed voltage to variable frequency variable voltage and is fed to
the single phase induction motor. The actual speed as sensed by the sensor
and the reference speed are compared by the comparator and error in the
speed is obtained. The current and voltage controllers are used to generate
pulses to the matrix converter. SPWM is used as the control strategy to
regulate the speed of single phase induction motor.
135
Figure 6.1 Block diagram of single phase matrix converter fed
induction motor
Figure 6.2 Schematic representation of single phase matrix converter
The main circuit of a matrix converter is composed of an input
filter bidirectional switches also known as bilateral switches. Mostly two anti-
parallel reverse blocking IGBT’s are used as bidirectional switches. In
addition to bidirectional operation with the same set of switches, an optimized
power factor, reduced harmonic content of input currents and voltage stress
are also achieved (Sunter and Aydogmus 2008).
6.1.2 Modulation Strategy
To obtain an output voltage of desired magnitude and frequency,
the matrix converter chops the high frequency input AC voltage. SPWM
136
technique is used to generate switching signals for the switches S2 and S4
respectively while the switches S1 and S3 are operated at modulation
frequency, fm. The fundamental component of the output voltage is given by
Equation (6.1)
imo fff (6.1)
where mf is the modulation frequency and if is the input frequency. The
output voltage is proportional to the modulation ratio. The input voltage is
given by Equation (6.2)
iV (t) = imV cos i t (6.2)
The output voltage will be Equation (6.3)
tVo omV cos o t (6.3)
To control the power switches S1, S2, S3 and S4, the following four
switching states are used. The other possible states are not applied (Sunter
and Aydogmus 2008).
The first state being the ‘ON’ state for S1 and S4 will be
during the positive cycle with the output voltage as the input
voltage.
The negative cycle of input voltage will be the second state
keeping the switches S2 and S3 in 'ON' state.
The remaining two states will have zero output with
switches S1 and S2, S3 and S4 are in ‘ON’ state respectively.
137
6.1.3 System Modeling
The simulation model of closed loop control of the proposed
system driving a single phase induction motor with the reference speed and
the applied voltage is given by Figure 6.4.
Figure 6.3 MATLAB / SIMULINK Model of the proposed system
without feedback
Figure 6.4 MATLAB / SIMULINK Model of the proposed system with
feedback
138
6.1.4 Simulation Results and Discussion
An input voltage of magnitude 150 V and frequency 50 Hz is
applied to the SPWM based single phase matrix converter and the results are
analyzed. The various simulation results are displayed in Figures 6.5 to 6.8.
Figure 6.5 Input voltage Vs Time
Figure 6.6 SPWM signals
The modulation scheme and switching signals of the proposed
system is shown in Figure 6.6. When a carrier signal with a frequency of 1.25
kHz and reference sinusoidal voltage of frequency 50 Hz are compared with
the relational operator and logical operator, SPWM pulses with variable pulse
width are generated. These pulses are used to control the output voltage and
Inp
ut
Vo
lta
ge
(V)
Time (s)
Vm,
Vc
VS4
VS2
Time (s)
Vm Vc
139
output current of the matrix converter. The output voltage of 150 V and a
current of 0.15 A obtained are shown in Figures 6.7 and 6.8 respectively.
Further study of harmonic spectrum in the output voltage signal using FFT
analysis yields a Total Harmonic Distortion (THD) of 15.95% as given by the
Figure 6.9.
Figure 6.7 Output voltage Vs Time Figure 6.8 Load current Vs Time
Figure 6.9 Harmonic spectrum of output voltage
To drive a ½ HP, 230 V, 50 Hz, 1.4 A, single phase induction
motor fed by SPWM based matrix converter, a closed loop control to regulate
the speed of the motor using PID controller is employed. The simulation
Time (s)
Ou
tpu
t V
olt
ag
e (V
)
Lo
ad
Cu
rren
t (A
)Time (s)
140
results are shown in Figures 6.10 to 6.12. The input voltage of magnitude 230
V and frequency 50 Hz is give as input. The matrix converter converts fixed
frequency fixed voltage to variable frequency variable voltage to obtain the
required speed for the motor. An out voltage of 225 peak to peak is obtained
for a modulation depth of 0.75.
Figure 6.10 Input voltage Vs Time
Figure 6.11 Output voltage Vs Time Figure 6.12 Speed Vs Time
Figure 6.12 shows the rotor speed of single phase induction motor.
The speed gradually increases from zero reaches 1480 rpm in 2 sec. The
transient state exists till 2.3 sec and it reaches its steady state value of 1400
rpm. The total harmonic distortion in output voltage waveform is found to be
19.75 % from the Figure 6.13.
Inp
ut
Vo
lta
ge
(V)
Time (s)
Ou
tpu
t V
olt
ag
e (V
)
Time (s) Time (s)
Mo
tor S
pee
d (
RP
M)
141
Figure 6.13 Harmonic spectrum of output voltage
6.1.5 Model Validation
For experimental studies, a laboratory model of single phase matrix
converter has been designed to feed various types of loads such as R-load, R-
L load or a single phase induction motor. Square wave PWM pulses have
been generated using a dsPIC 30F3011 microcontroller. The switching
frequency was taken as 10 kHz, and the modulation frequency was varied in
steps from 0 to 50 Hz. The modulation depth was varied between 0.01 and 1.
The output pulses from the microcontroller are given to gate driver circuit of
the converter switches.
The matrix converter is capable of blocking voltage and conducting
current in both directions. As the bidirectional switches are not available in
the market today, additional construction of the same with some power
semiconductor devices is needed. Supertex GN2470 is a 700 V, 3.5 A
Insulated Gate Bipolar Transistor (IGBT) that combines the positive aspects
142
of both Bipolar Junction Transistors (BJT) and Metal Oxide Field Effect
Transistors (MOSFET) is used. This device possesses lower on-state voltage
drop with high blocking voltage capabilities and low conduction voltage drop
at high currents. It is not possible to simply turn off all the switches as this
will open circuit the inductive motor load causing a very high voltage
transients. Hence a small capacitor is used for this clamp and it is rated to take
the energy stored in the inductance of the load without exceeding the
maximum voltage rating of the devices.
Figure 6.14 Block diagram of hardware model
The block diagram of the hardware model of single phase AC-AC
converter fed induction motor drive is shown in Figure 6.14. Gate driver
circuit is designed in such a way to detect collector-to-emitter voltage ceV of
a turned-on IGBT, short circuits etc. In square wave PWM, the gating signals
are generated by comparing a square wave reference signal of frequency rf
with a triangular carrier wave of frequency cf . The reference frequency
determines the output frequency of and its peak amplitude controls the
modulation index and in turn the RMS output voltage. A 40 pin dsPIC
30F3011 microcontroller is used as pulse generator. To obtain four different
constant voltages (+5V, +10V, +12V, -5V), switched mode power supply is
used. The circuit diagram of SMPS is shown in Figure 6.15. It consists of
143
pulse generator, MOSFET (2SK2848) and high frequency pulse transformer,
chopper, opto-coupler (PC923) and filter.
Figure 6.15 Circuit diagram of SMPS
The hardware of matrix converter as shown in Figure 6.16 was
fabricated and the output is fed to the drive. An R-L Load of (R = 10 ,
L = 0.1 H) is connected across the matrix converter. As shown in Figure 6.17
(a), a single phase AC voltage of 115 V and frequency 50 Hz is given as input
to the converter. The microcontroller is used to produce control signals based
on square wave pulse width modulation technique for the gates of the IGBT.
The hardware results are analyzed under various Voltage / Frequency ratios.
The output waveforms and the harmonic spectra are measured using Power
Quality Analyzer and presented in the following figures. From Figure 6.17
(b), the input current is about 1 A.
144
Figure 6.16 Experimental set-up
The output voltage shown in Figure 6.18 (a) is of pulsed sinusoidal
waveform. The peak to peak voltage is 114 V and frequency is 50 Hz. Figure
6.18 (b) depicts the output current waveform at 50 Hz frequency.
(a) (b)
Figure 6.17(a, b) Input voltage and current waveforms
145
(a) (b)
Figure 6.18(a, b) Output voltage and current for 0f 50 Hz with R-L load
Now a fan load of capacity ½ HP, 220 V, 1.4 A is connected across
the matrix converter. The output voltage shown in Figure 6.19 (a) is of pulsed
sinusoidal waveform. The peak to peak voltage is 91 V and frequency is 45
Hz. The output current is approximately 1 A at 45 Hz.
(a) (b)
Figure 6.19(a, b) Induction motor voltage and current for 0f 45 Hz
Figure 6.20 Harmonic spectrum of output voltage for 0f 45 Hz
146
The THD level of output voltage at frequency 0f 45 Hz is 59.8 % of the
fundamental frequency as given by the Figure 6.20. Form the harmonics table
it is observed that harmonics present in output voltage waveform of third
order is 33.0 %, fifth order is 4.4 %, seventh order is 2.5 %, ninth order is 0.7
%, eleventh order is 2.4 %, thirteenth order is 2.4 % and fifteenth order is 5.5
% of fundamental frequency respectively. For various frequencies, the results
are obtained and shown in Figures 6.21 – 6.22. It is observed, that the rms
voltage and peak voltage of matrix converter are 101.7 V and 167.2 V
respectively for 48 Hz. The peak current of matrix converter is 1A.
(a) (b) (c)
Figure 6.21(a, b, c) Output voltage, current and harmonic spectrum for
0f 48 Hz
(a) (b) (c)
Figure 6.22(a, b, c) Output voltage, current and harmonic spectrum for
0f 50 Hz
147
For 90 V / 45 Hz, the total harmonic distortion is 60 % and for 115
V / 50 Hz the total harmonic distortion is 36 %. When the output frequency is
equal to supply frequency, the total harmonic distortion is low. As the output
voltage increases the harmonics also increases.
Figure 6.23 Speed response of the induction motor
Figure 6.23 shows the speed characteristics of the single phase
induction motor without feedback. For the set value of frequency, the V/f
ratio of the drive is automatically maintained and it runs at its nearest
synchronous speed. The motor takes 6.3 sec to reach its steady state value of
1410 rpm, when compared to the steady state speed of 1400 rpm in the
simulation study. Thus the model is validated.
6.3 THREE PHASE MATRIX CONVERTER
In literature, several methods of incorporating conventional AC-
DC-AC conversion system for wind power applications have been discussed.
Usually the generator side converter will be a controlled / uncontrolled
rectifier or a voltage source converter (VSC). The utility side converter is
often a VSC unit. The main drawbacks of this type of conversion are
increased size and weight, low reliability of DC link capacitor; poor line
power factor and significant harmonic distortion in line and machine currents
(Vinod Kumar et al 2009). The IEEE Standard 519 (1991) severely restricts
148
line harmonic injection. Matrix converters will avoid the conventional AC-
DC-AC conversion, so the steps involved in the conversion is reduced. Due to
the lack of DC link capacitor for energy storage, the total size of the system is
reduced. The following sections deal with the three phase matrix converters
applied for wind turbine generators.
6.4 CONVERTER TOPOLOGIES
The different converter topologies are discussed below.
6.4.1 Indirect Matrix Converter
The indirect matrix converter topology consists of 18 IGBT
switches with an extra leg as shown in Figure 6.24. So this converter has 34
switching states. The switching combinations are same as that of any matrix
converter except some extra switching combinations. This topology is more
complex but easy to control. The modulation can be divided into two, input
side modulation and the load side modulation. In order to have better control,
input modulation method is developed. The switch used is IGBT because of
high power handling capability.
Figure 6.24 Schematic representation of indirect matrix converter
149
6.4.2 Nine Switched Matrix Converter
As mentioned by Ebubekir Erdem et al (2005), conventional way of
matrix converter concept is nine switch topology using bi-directional
switches. They are arranged as three sets of three, so that any of the three
input phases can be connected to any of the three output lines, as shown in
Figure 6.25. The matrix components S11, S12, …, S33 represent nine bi-
directional switches which are capable of blocking voltage in both directions
and of switching without any delays (Alberto Alesina and Marco Venturini,
1989). The matrix converter connects the three given inputs, with constant
amplitude, Vi and frequency, fi through the nine switches to the output
terminals in accordance with pre calculated switching angles. The three-phase
output voltages obtained have controllable amplitudes, Vo and frequency, fo.
The number of switching states in the matrix converter is 27 (33) and the
switching states can be achieved by changing sequence of the signal given to
the switches. The switch used is IGBT because of high power handling
capability.
Figure 6.25 Schematic representation of nine switched matrix converter
IGBT Switch
150
6.5 MODELING OF VENTURINI ALGORITHM BASED
MATRIX CONVERTER
A Venturini’s modulation algorithm is used for matrix converter
and is defined in terms of the three phase input and output voltages at each
sampling instant and can be easily implemented for closed loop operations
(Altun and Sunter, 2003). This algorithm also provides unity fundamental
displacement factor at the input regardless of the load displacement factor. In
this method, a set of three-phase input voltages with constant amplitude and
frequency is considered, for calculating the duty cycle of each of the nine
bidirectional switches. The result thus obtained and when implemented allows
the generation of a set of three-phase output voltages by sequential piecewise
sampling of the input waveforms (Zhang et al 1998).
A simplified version of the Venturini algorithm as discussed by
Sunter (1995) is used here to simulate and analyze the performance of the
matrix converter. For the real-time implementation of the proposed
modulation algorithm, it is required to measure any two of three input line-to-
line voltages. Then, input peak voltage Vim and and input position it are
calculated as shown in Equations (6.4) and (6.5). Similarly the target output
peak voltage and the output position can be calculated as Equations (6.6) and
(6.7), where VAB, VBC are the instantaneous input line voltages and va, vb, vc
are the target phase output voltages.
BCABBCABim VVVVV222
94 (6.4)
)3
13
2(3arctan
BCAB
BCit
VV
V (6.5)
cbaom vvvV9
4 (6.6)
151
)(3arctan
a
cbot
v
vv (6.7)
Alternatively, in a closed loop system the voltage magnitude and angle may
be the direct outputs of the control loop. Then, the voltage ratio is calculated
from Equation (6.8).
im
om
V
Vq (6.8)
where q is the desired voltage ratio, and Vim is the peak input voltage.
Triple harmonic terms are found as Equations (6.9) – (6.11)
)3sin()sin(94
31 ititqmqk (6.9)
)3sin()sin( 32
94
32 ititqmqk (6.10)
)3cos()3cos( 14
16
133 itqmotomVk (6.11)
where qm is the maximum voltage ratio (0.866).
Then, the three modulation functions for output of phase ‘a’ are given as
Equations (6.12) – (6.14).
BCABimVVaVAa kvkM
3
1
3
2
333
2
3194 )( (6.12)
ABBCaimBa VVkvVkM3
1
3
1)(
3
23
13332 (6.13)
)(1 BaAaCa MMM (6.14)
152
The modulation functions for the other two output phases, b and c
are obtained by replacing vb and vc with va, respectively in Equation (6.12)
and Equation (6.13). The modulation functions have third harmonic
components at the input and output frequencies added to them to produce
output voltage, vo. This is a requirement to get the maximum possible voltage
ratio. It is noted that in Equation (6.6) that there is no requirement for the
target outputs to be sinusoidal. In general, three phase output voltages and
input currents can be defined in terms of the modulation functions in matrix
form as given by Equation (6.15) and Equation (6.16).
Output voltage per phase = M function*Input voltage per phase
C
B
A
CcBcAc
CbBbAb
CaBaAa
c
b
a
v
v
v
MMM
MMM
MMM
v
v
v
(6.15)
Input current per phase = Transpose of M function*Output current per phase
c
b
a
CcCbCa
BcBbBa
AcAbAa
C
B
A
i
i
i
MMM
MMM
MMM
i
i
i
(6.16)
where M is the instantaneous input-phase to output-phase transfer matrix of
the three-phase matrix converter. viph and voph are the input and output phase
voltage vectors and iiph and ioph represent the input and output phase current
vectors. Alternatively, from Equation (6.15) and Equation (6.16), the output
line voltages and input line currents can be expressed as Equation (6.17) and
Equation (6.18) respectively.
Output line voltage = m-function*Input line voltage
153
CA
BC
AB
CaBaAa
CcBcAc
CbBbAb
ca
bc
ab
V
V
V
mmm
mmm
mmm
v
v
v
(6.17)
Input line current = Transpose of m-function*Output line current
ca
bc
ab
caCcCb
BaBcBb
AaAcAb
CA
BC
AB
i
i
i
mmm
mmm
mmm
I
I
I
(6.18)
where Aam , Abm and Acm are calculated displaced vectors of AaM , AbM
and AcM respectively. Similarly modulation values of other two phases are
also calculated as given in Equation (6.19):
)()(
)()(
)()(
)()(
)()(
)()(
)()(
)()(
)()(
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
AaAcCaCcCa
CaCcBaBcBa
BaBcAaAcAa
AcAbCcCbCc
CcCbBcBbBc
BcBbAcAbAc
AbAaCbCaCb
CbCaBbBaBb
BbBaAbAaAb
MMMMm
MMMMm
MMMMm
MMMMm
MMMMm
MMMMm
MMMMm
MMMMm
MMMMm
(6.19)
Figure 6.26 shows the model of the matrix converter in blocks.
Ideal switches have been assumed in the simulation of the matrix converter
power circuit. The input variables to the matrix converter are the clock, input
voltages (Vi) and target output voltages (Vo) obtained from a controller. The
target voltage is given as second input (Vo). The input voltage block
represents Equations (6.4) and (6.5). The target output block represents
Equations (6.6) and (6.7). The M function block, represents Equations (6.12) -
154
(6.14) for single phase and similar three phase calculations are performed.
Modulation functions are taken out for calculating the output voltages and
input currents using Equation (6.17) and Equation (6.18) respectively. The
switching frequency of the matrix converter is determined by the signal
generator block, where fs = 2 kHz.
The block ‘‘Duty cycle generator’’ consist of logic gates and
simple mathematical algorithms. Here the modulation functions are compared
with the signal generator waveform, probably square or saw tooth at the input
and arranged to have logic levels. Logic gates are used at the output to get
three gate signals proportional to the duty cycle of the power switches for one
output phase. The ‘‘switch’’ option is a SIMULINK building block. It has
three inputs, in(1), in(2) and in(3) and one output. It operates in accordance
with the following logic:
if in(2)>0 then output=in(1)
else output=in(3)
Similarly nine switches are used to have three phase output voltages.
Figure 6.26 Block schematic of matrix converter model
Clock
Input
Voltage
Target
Voltage
M
Functions
Duty Cycle
Block
Signal
Generator
Nine
Switches
For calculating
currents and voltages
Output
155
6.6 MODELING OF SPACE VECTOR MODULATED MATRIX
CONVERTER
Space Vector Modulation (SVM) is an algorithm similar to the
rotating flux of the induction machine. This is based on the representation of
the three phase input currents and three phase output line voltages on the
space vector plane. SVM completely exploit the possibility of matrix
converters to
control the input power factor regardless the output power factor
fully utilize the input voltages
reduce the number of switch commutations in each cycle period
Since the matrix converter is supplied by the wind turbine driven self-excited
induction generator, the input phases must never be short-circuited and as the
loads are inductive in nature, the output phases must not be left open-
circuited.
Based on the above two conditions, only 27 (33) switching
combinations are possible out of 512 (29). These can be classified into three
groups with 6, 18 and 3 different switching combinations respectively. In the
first group of 6 combinations, each output phase is connected to a different
input phase. The second group of 18 combinations has two output phases
short-circuited. These are the active configurations that determine an output
voltage vector and an input current vector, having fixed directions. The third
group includes 3 combinations with all the three output phases short-circuited.
They are the zero configurations that determine zero input current and zero
output voltage vectors (Ebubekir Erdem et al 2005 and Casadei et al 1993).
The three phase waveforms are converted to d-q co-ordinates using
Park’s transformation method (Alexandru, 2007) as given by Equation (6.20).
156
2
1
2
1
2
1
)3/4sin()3/2sin(sin
)3/4cos()3/2cos(cos
3/2)( (6.20)
For = 0, Clarke’s transformation matrix is obtained. The Concordia matrix
of abc- transformation is given by Equation (6.21).
2/12/12/1
232/30
2
1
2
11
3/2][Cc (6.21)
The multiplication of Va,Vb, Vc with [Cc] gives the values V and V as
Equations (6.22) and (6.23):
cba VVVV2
1
2
1816.0 (6.22)
)866.0866.0(816.0 cb VVV (6.23)
Figure 6.27 Space vector
representation of output
line voltages
Figure 6.28 Space vector
representation of input
currents
157
The rotating vector is divided into 6 sectors in which each sector
has a phase difference of 60o. The sectors are divided as 1, 2, 3, 4, 5 and 6 in
the way 0o - 60
o, 60
o - 120
o, 120
o - 180
o, 180
o - 240
o, 240
o - 300
o, 300
o - 360
o
respectively. Based on the sectors, the d-q coordinates are determined.
The switching function for each switch is,
Skj= {1, Switch Skj closed
{0, Switch Skj opened
where K= {A,B,C}, j={a,b,c}
SKa + SKb + SKb = 1
The mathematical relationship between input and output instantaneous phase
voltages is given by Equation (6.24),
CcCbCa
CcBcBa
AcAbAa
C
B
A
SSS
SSS
SSS
v
v
c
b
a
v
v
v
(6.24)
where vA, vB, vC and va, vb, vc represent input and output phase voltages.
Based on the above equation, line-line voltages and phase currents at the
output and input terminals are given by Equation (6.25) and Equation (6.26).
C
B
A
AcCcAbCbAaCa
CcBcCbBbCaBa
BcAcBbAbBaAa
CA
BC
AB
v
v
v
SSSSSS
SSSSSS
SSSSSS
v
v
v
(6.25)
C
B
A
CcBc
CbBbBa
CcBaAa
C
B
A
i
i
i
SSSAc
SSS
SSS
i
i
i
(6.26)
where ABv , BCv , CAv and Ai , Bi , Ci are output voltages and currents respectively.
158
The instantaneous input source voltages are given by Equations (6.27) –
(6.29)
Av = imv sin it (6.27)
Bv = imv sin ( it-2 /3) (6.28)
Cv = imv sin( it+2 /3) (6.29)
Figure 6.27 shows the sector representing output voltages, each
vector representing the output line voltages. The sectors are such that they are
at a phase difference of 60o. The space vector representation of input current
is illustrated in Figure 6.28. The input phase voltages to the matrix converter
as expressed in Equation (6.30) are at a phase difference of 120oelectrical.
3
2cos(
)3
2cos(
)cos(
t
t
t
V
i
i
i
im
c
b
a
(6.30)
The reference phase voltages are given by Equation (6.31)
)3
2
6cos(
)3
2
6cos(
)6
cos(
3
oo
oo
oo
om
CA
BC
AB
t
t
t
V (6.31)
The output voltages are based on the reference phase voltages Equation
(6.31), given to the converter.
159
The output currents are given by Equation (6.32)
)3
2cos(
)3
2cos(
)cos(
Loo
Loo
Loo
om
C
B
A
t
t
t
I
i
i
i
(6.32)
The reference input currents are given by Equation (6.33)
)3
2cos(
)3
2cos(
)cos(
ii
ii
ii
im
c
b
a
t
t
t
I
i
i
i
(6.33)
The reference input currents determine the output waveforms. The
switching of matrix converter depends upon the amplitude and the frequency
of the reference input. The switching combinations are determined by the
order of selected sequence. The frequency of reference voltage and frequency
of carrier wave determines the switching pulse of the SVM. The switching
sequence is defined by the order which is selected from the total number of
switching combinations, 27.
The detailed switching combinations in six sectors with respect to
input voltage and input current are tabulated in Table 6.1. So there will be 36
(6 x 6) sectors considering all the switching combinations. Here only one
sequence of switching combination is switching is used. For example, in
combination 1, +1 means forward switch of first group (out of nine groups)
and -3 means reverse switch of third group and so on (Ebubekir Erdem et al
2005).
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Table 6.1 Switching combinations in six sectors for matrix converter
iI
oV1 2 3 4 5 6
1+1-3
-4+6
+2-3
-5+6
-1+2
+4-5
-1+3
+4-6
-2+3
+5-6
+1-2
-4+5
2-7+9
+1-3
-8+9
+2-3
+7-8
-1+2
+7-9
-1+3
+8-9
-2+3
-7+8
+1-2
3+4-6
-7+9
-8+9
+5-6
-4+5
+7-8
-4+6
+7-9
-5+6
+8-9
+4-5
-7+8
4+4-6
-1+3
5-6
-2+3
-4+5
+1-2
+1-3
-4+6
+2-3
-5+6
+4-5
-1+2
5-1+3
+7-9
-2+3
+8-9
+1-2
-7+8
+1-3
-7+9
+2-3
-8+9
-1+2
+7-8
6+7-9
-4+6
+8-9
-5+6
-7+8
+4-5
+4-6
-7+9
+5-6
-8+9
+7-8
-4+5
Table 6.2 Switching configurations
Configuration ABC Configuration ABC
+1 abb 1 -4 aba
-3 acc 2 +1 abb
-4 aba 3 -3 acc
+6 aca 4 +6 aca
0 aaa 5 0 aaa
0 bbb 1 -4 aba
0 bbb … … …
Table 6.2 shows the switching configurations generally used in the
matrix converter nine switched topology. Here A, B, C, and a, b, c represent
the output phases and input phases respectively. To have three phase voltage
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transfer, any of switches of all the three phases must be in ON position while
the switches in the same leg must not be switched ON simultaneously to
avoid short-circuit between the switches. Hence three switching combinations
are invalid vectors (Ebubekir Erdem et al 2005 and Casadei et al 1993).
6.7 SIMULATION RESULTS AND DISCUSSION
6.7.1 Matrix Converter employing Venturini Algorithm
The matrix converter described in section 6.5 is modeled and
simulated using MATLAB / SIMULINK for both loaded and unloaded
conditions. The simulation diagrams for both cases are depicted in
Figures 6.29 and 6.30 respectively. The results obtained are analyzed as
follows. The parameter inputs under unloaded and loaded conditions are
tabulated in Table 6.3.
Figure 6.29 MATLAB / SIMULINK model of a three phase matrix
converter under unloaded condition
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Table 6.3 Operating conditions used for analysis
Parameter inputs for 3-phase matrix converter under unloaded
condition
Input specifications:
Input voltage, Vi = 311.08 V
Input frequency, fi = 50 Hz
Vo/Vi = 50 %
Output target specifications:
Output voltage, Vo = 155.54 V
Output frequency, fo = 100 Hz
Switching frequency, fs = 2000 Hz
Parameter inputs for 3-phase matrix converter under loaded condition
Input specifications:
Input voltage, Vi = 311.08 V
Input frequency, fi = 50 Hz
Vo/Vi = 50 %
Load specifications:
R = 20 ; L = 0.04 Henry
Output target specifications:
Output voltage, Vo = 155.54 V
Output frequency, fo = 100 Hz
Duty cycle frequency, fs = 2000 Hz
Figure 6.30 MATLAB / SIMULINK model of a three phase matrix
converter under loaded condition
163
Figure 6.31 Input voltage waveform of three phase matrix converter
Figure 6.32 Output voltages of phase a, b and c under unloaded condition
Figure 6.33 Waveforms of calculated M-functions for one phase
164
Figure 6.34 Waveforms of calculated M-functions for all the three phases
Figure 6.35 Load current for one phase
165
Figure 6.36 Load current for all the three phases
Figure 6.37 FFT Analysis of load current for one phase
166
The simulation results of unloaded matrix converter operated at the
specified parameters in Table 6.3 are presented in Figures 6.31 – 6.37. Figure
6.31 represents the input voltage waveform with a magnitude of 311.08 V.
The simulated results of the three output voltages are illustrated in Figure
6.32. Figure 6.33 represents the calculated M-function values for one phase,
say “a”. Calculated M-values for all the three phases are given in Figure 6.34.
At this stage one can summarize that the pre calculated M-values provide high
quality switching pattern.
Now the three phase matrix converter is loaded with star connected
three phase R-L load (20 and 0.04 Henry) as shown in Figure 6.29. Here
unity input power factor is considered, which means one phase input current
is low passed compared with the respective input voltages. The simulation
results of loaded matrix converter are presented in Figures 6.35-6.37. The
results of load current for one phase and for all the three phases are presented
in Figures 6.35 and 6.36 respectively. The load currents are nearly sinusoidal.
It can be further improved by modulating the switching frequency.
Figure 6.37 shows their FFT analysis. The fundamental harmonics
is at 50 Hz and the additional harmonics are around the switching frequency.
The Total Harmonic Distortion (THD) is found to be 17.74 % of the
fundamental frequency of 50 Hz.
6.7.2 Conventional AC-DC-AC Converter using Space Vector
Modulation
Figure 6.38 shows MATLAB / SIMULINK model of conventional
AC-DC-AC converter employed with SVM. It consists of 12 IGBT switches.
The SVM pulses are given as the firing signals to the IGBTs. They are shown
for six sectors in Figure 6.39. According to the given sequence the pulse is
generated. The sequence is 0o-60
o, 60
o-120
o, 120
o-180
o, 180
o-240
o, 240
o-300
o,
167
300o-360
o. Each sequence represents single sector in the order s1, s2, s3, s4, s5
and s6 respectively. The simulation parameters are tabulated in Table 6.4.
Figure 6.38 Simulation model of conventional SVM based AC-DC-AC
Converter
Figure 6.39 SVM pulses
168
Table 6.4 Simulation parameters of converters
Input Voltage Vin 400 V (Line Voltage)
Input Frequency fin 50 Hz
Switching Frequency fs 5000 Hz
Switching Period Ts 20 µsec
R-Load 2000
L ( R-L Load ) 1.5 H
Figure 6.40 Input line voltages to the converter
Figures 6.40 to 6.42 show the simulation results of the conventional
AC-DC-AC converter. Due to losses in the two conversion stages, the output
line voltage is 350 V. FFT analysis of the output voltage and output current is
done using MATLAB and the harmonic spectrum of both cases are presented
in Figure 6.43.
(a) (b)
Figure 6.41(a, b) Output line and phase voltages for R-Load
169
From the results, it is seen that THD level is 20.45 % and 20.46 %
of the fundamental frequency of 50 Hz respectively. The harmonic injection is
due to the DC link present in the model. The simulation results are
summarized in Table 6.5.
(a) (b)
Figure 6.42(a, b) Output current for R-Load and R-L Load
(a) (b)
Figure 6.43(a, b) Harmonic spectrum of output voltage and current
Table 6.5 Simulation results of AC-DC-AC converter
Vo (Output Voltage)230 V (Phase Value)
350 V (Line Value)
Io (Output Current for R-Load)
Io (Output Current for R-L Load)
0.125 A
0.099 A
THD (for Voltage)
THD (for Current)
20.45 %
20.46 %
170
6.7.3 Indirect Matrix Converter using Space Vector Modulation
The indirect Matrix Converter is an advanced version of AC-DC-
AC converter. This model does not contain the DC link but it has two parts
separated as input and output. The simulation model is illustrated in Figure
6.44. There are 18 switches and separate legs for input and output
respectively. The switching is done using SVM pulses. Here the SVM pulses
are given to the gates according to the switching order. The extra leg in the
circuit gives more number of switching combinations.
Figure 6.44 Simulation model of indirect matrix converter
(a) (b)
Figure 6.45(a, b) Output line and phase voltages for R-Load
171
(a) (b)
Figure 6.46(a, b) Output current for R-Load and R-L Load
(a) (b)
Figure 6.47(a, b) Harmonic spectrum of output voltage and current
The simulation results are depicted in Figures 6.45 and 6.46. Figure
6.45 shows the output AC voltages for R-Load. There is an intermediate
fictitious DC link and hence the harmonic distortion is reduced. The results
obtained from FFT analysis is shown in Figure 6.47. Table 6.6 summarizes
the results obtained for both R and R-L loads.
Table 6.6 Simulation Results of Indirect Matrix Converter
Vo (Output Voltage)230 V (Phase Value)
350 V (Line Value)
Io (Output Current for R-Load)
Io (Output Current for R-L Load)
0.125 A
0.099 A
THD (for Voltage)
THD (for Current)
20.38 %
20.38 %
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6.7.4 Nine Switched Matrix Converter using Space Vector Modulation
Figure 6.48 illustrates the simulation model of the direct AC-AC
converter with nice switch topology. There are 27 switching combinations.
The avoidance of intermediate step causes reduction in losses. The switching
is done using SVM pulses. The simulation results are shown in Figures 6.49
and 6.50.
Figure 6.48 Simulation model of matrix converter (Nine switch topology)
(a) (b)
Figure 6.49(a, b) Output line and phase voltages for R-Load
173
(a) (b)
Figure 6.50(a, b) Output current for R-Load and R-L Load
(a) (b)
Figure 6.51(a, b) FFT Analysis of output voltage and current
The percentage of THD as shown by Figure 6.51 indicate that the
performance of proposed model when compared to the conventional converter
with 5.11 % and 5.12 % respectively. The entire results obtained are tabulated
in Table 6.7. FFT analysis shows that the performance of improved matrix
converter is better than the conventional converter but not impressive as
matrix converter. The comparison between converters as summarized in Table
6.8 shows that matrix converter (nine switch topology) is more efficient in
reducing harmonics and voltage transfer.
174
Table 6.7 Simulation results of nine switched matrix converter
Vo (Output Voltage)230 V (Phase Value)
400 V (Line Value)
Io (Output Current for R-Load)
Io (Output Current for R-L Load)
0.125 A
0.099 A
THD (for Voltage)
THD (for current)
5.11 %
5.12 %
Table 6.8 Comparison of power electronic converters
ParametersAC-DC-AC
Converter
Indirect matrix
Converter
Nine Switched
Matrix
converter
Switching
speedLow High High
Voltage
transfer ratio400 V / 350 V 400 V/ 350 V 400 V/ 400 V
THD %
(V/I)20.46 / 20.45 20.38 / 20.38 5.11 / 5.12
Frequency
regulationNot possible Possible Possible
The simulation model of closed loop speed control of SVM based
three phase matrix converter feeding a three phase induction motor is shown
in Figure 6.52. The control consists of voltage control loop and current
control loop. For continuous change in the reference speed, the output is
controlled at the desired level. The control loops will sense the state of the
system through a reference block and so the voltage level of the system is
maintained. The simulation parameters along with the details of induction
motor load are given in Table 6.9. The simulation results of closed loop
system to control the speed of the induction motor are shown in Figures 6.53.
For a set speed of 157 rad/sec, the electromagnetic torque, output currents and
175
the rotor speed are shown in Figure 6.53(c). Irrespective of the input voltage,
the speed is maintained constant. This illustrates the effectiveness of the
control.
Figure 6.52 Simulation model of closed loop control of matrix converter
Table 6.9 Simulation parameters of closed loop operation
Input Voltage
Vin
400 V
(Line)
Input
Frequency fin
50 Hz
Switching
Frequency fs
5000
Hz
Ts 20 µs
Load Type Induction Motor
Capacity 5 HP
Rated Voltage 400 V
Rated Current 7.5 A
Rs (Stator Resistance)
Ls (Stator Inductance)
0.087
0.8 mH
Rr (Rotor Resistance)
Lr (Stator Inductance)
0.228
0.8 mH
Lm 34.7 mH
Operating Frequency 50 Hz
Rated Speed 1500 rpm
176
(a)
(b)
(c)
Figure 6.53(a, b, c) Simulation results of closed loop control
177
Figure 6.53(d) Output voltage
6.8 HARDWARE IMPLEMENTATION
The block schematic of the hardware set-up of a three phase matrix
converter is shown in Figure 6.54. It consists of power module, control
module and required protection circuits. The specifications are given in
Appendix.
FPGA
Figure 6.54 Block schematic of hardware set-up
178
The design and implementation of matrix converter circuit along
with the required modules is done by considering the current state of
semiconductor technology and the power rating of the prototype model.
6.8.1 Power Module
The power module consists of nine bidirectional switches capable
of blocking voltage and conducting current in both directions. These switches
are made up of 9 bidirectional switches with NPT3 IGBT and fast diode
bridge (FIO 50-12BD) for the reverse flow of current. The functional circuit
diagram of the power switches is shown in Figure 6.55.
Figure 6.55 Functional circuit diagram of matrix converter
Input filter is used to reduce the switching frequency harmonics
present in the input current. Simple LC filter is used for this purpose. The
requirements of the LC filter are,
Cut-off frequency lower than the switching frequency of matrix
converter
Minimal reactive power at grid frequency
Minimal volume and weight
Minimal filter inductance voltage drop at rated current in order
to avoid reduction in voltage reduction ratio
179
The filter is designed in such a way that the input and output
currents of the converter are not affected.
Isolated gate driver (HCPL-4506) circuits as given in Figure 6.56
are used to amplify the controller voltage of 5 V to the switching voltage of
15 V for the power switches. It contains a GaAsP LED which is optically
coupled to an integrated high gain photo detector. Due to minimized
propagation delay difference between the devices, these optocouplers improve
the inverter efficiency through reduced switching dead time.
Figure 6.56 Driver circuit for IGBTs
6.8.2 Control Module
Controller is the heart of the whole module. Matrix converters
require complex control algorithm and hence the programming is
implemented using Field Programmable Gate Arrays (FPGA). FPGA are
arrays of logic blocks, surrounded by programmable I/O blocks which can be
linked together to form complex logic implementations. A typical FPGA
180
contains from 64 to tens of thousands of logic blocks and an even greater
number of flip-flops. The individual cells are interconnected by a matrix of
wires and programmable switches. The array of these logic cells and
interconnections form a fabric of basic building blocks for logic circuits.
Complex designs are created by combining these basic blocks to create the
desired circuit. The FPGA technology offers a many more advantages as
compared to other conventional technologies. The program can easily be
changed to optimize and control the converter parameters like frequency and
voltage. A very attractive high-level simulation tool is provided by FPGA,
called XILINX. (Prawin Ange et al 2011).
Xilinx Spartan-3E XC3S100E FPGA is a 144-Thin Quad Flat Pack
package which is used to generate gating signals. After initializing the control
board, VHDL (Very High Speed Integrated Chip Hardware Description
Language) code of PWM signal generation using Xilinx ISE 10.1 software is
written as per the logic given below.
PWM is a technique to provide a logic “1" and logic “0" for a
controlled period of time. Zero crossing detected three phase voltage is
compared with the sinusoidal PWM signal generated from three phase
reference voltages. Thus nine pulses to the matrix converter switches are
generated. The pulse pattern for the bi-directional switches is shown in
Table 6.10.
Table 6.10 Switching pulse pattern
Supply Voltage SPWM Voltage PWM Signal
RTOP V a top PWM1
YTOP V b top PWM2
BTOP V c top PWM3
RBOTTOM V a bottom PWM4
YBOTTOM V b bottom PWM5
BBOTTOM V c bottom PWM6
181
The PWM signal to the other switches is obtained by AND operation of
generated PWM signals as given by Equations (6.34) – (6.36).
(PWM1).(PWM4) = PWM7 (6.34)
(PWM2).(PWM5) = PWM8 (6.35)
(PWM3).(PWM6) = PWM9 (6.36)
PWM output is terminated in the box type header. Bidirectional voltage
translator (SN74LVCC3245A) is used in between FPGA I/O lines and box
type header to translate 3.3 V to 5 V and vice-versa.
6.8.3 Protection Circuits
In a matrix converter, over voltages and over currents can appear
from the input side. When the switches are turned OFF, the current in the load
is suddenly interrupted. The energy stored in the motor inductance (if load
used is motor) has to be discharged without creating over voltages. A clamp
circuit shown in Figure 6.57 is a common solution to avoid both input and
output over voltages. This clamp circuit has 12 fast –recovery diodes to
connect capacitor to input and output terminals.
Figure 6.57 Clamp circuit for over voltage protection
An isolation transformer with symmetrical windings is used to
decouple two circuits. It allows an AC signal or power to be taken from one
device and fed into another without electrically connecting two circuits. The
182
transmission of DC signals from one circuit to other is blocked while allowing
AC signals to pass. They also block interference due to ground loops. This
isolation transformer is used to give supply to ZCD (Zero Crossing Detector).
ZCD is used for comparing the phase signal with the reference signal. The
circuit diagram is shown in Figure 6.58.
Figure 6.58 Circuit diagram of zero crossing detector
183
6.8.4 Circuit Diagram
The overall circuit diagram of the prototype model is given
in Figure 6.59. The photographs of the experimental set-up are shown in
Figures 6.60(a, b).
Figure 6.59 Complete hardware circuit diagram
184
(a)
(b)
Figure 6.60(a, b) Snap shots of experimental set-up
6.8.5 Results and Discussion
The results of the experimental set-up are analyzed for both R-load
and a R-L load. V/f control is used to get a controlled output with required
185
frequency. The model validates the variable frequency output for different
load conditions. After setting the desired frequency, the input to the converter
is given from the supply mains or the from the prototype model of the wind
turbine driven self-excited induction generator. Also a single phase supply of
230 V / 9 V is given to the gate driver circuit. A variable resistive load is
connected across the converter.
(a) Input Voltage (fi = 50 Hz) (b) ZCD Output of R-Phase
(c) PWM Pulses to the Gate Circuit of One Leg
Figure 6.61 (Continued)
186
(d) Output Line Voltage (between R & Y) (fo = 50 Hz)
(e) Input Current (fi = 50 Hz) (f) Output Current (fo = 50 Hz)
Figure 6.61(a, b, c, d, e,f) Hardware results for 50 Hz frequency
Figures 6.61(a) – 6.61(f) illustrate the hardware outputs for an input
voltage of 110 V, 50 Hz frequency and the output voltage of 110 V, 50 Hz
frequency. The switching frequency is 10 kHz. It is seen that output
waveforms are almost sinusoidal and the THD level is found to be 7.9%
for the output voltage and 8.2% for the output current. The PWM
pulses generated is also given in Figure 6.61(c) for one leg of the converter
circuit. The output voltage and current for another resistive load is shown in
Figures 6.62.
187
Figure 6.62 Output voltage and current for varying resistive load
conditions
(a) Firing Pulses (b) Output phase voltages for 25 Hz
(c) Output Line voltages for 25 Hz (d) Output currents for 25 Hz
Figure 6.63(a, b, c, d) Hardware results for 25 Hz frequency
188
A variable R-L load is connected across the converter and the
results are observed for a set frequency of 25 Hz. The input line voltage of 60
V at a frequency of 50 Hz is given as input. The PWM pulses, the output R-
phase voltage and line voltage between R and Y phases, output current in R-
phase at 25 Hz are shown in Figures 6.63.The harmonic spectrum of the
output voltage is shown in Figure 6.64 where the THD level is 7.4 % of the
fundamental frequency of 50 Hz.
Figure 6.64 Harmonic spectrum of the output voltage
6.9 CONCLUSION
Matrix converters are reality and can be physically realized up to
100 kW in a single silicon structure. In this chapter, the concepts and
feasibilities of direct AC-AC matrix converter in wind turbine generators is
proposed. Matrix converter replaces the use of DC-link converters which
reduces losses produced due to the energy storage elements.
Single phase matrix converter feeding an R-L load and induction
motor has been simulated and implemented through hardware set-up.
Sinusoidal wave modulation algorithms have been used to see the input
voltage utilization and harmonic effects on the output. However, using the
189
sine waveform in the PWM algorithm, the amplitude of the output voltage has
increased. It is also found that using low frequency ratio, the lower order
harmonic contents increases in addition to higher amplitudes of the other
harmonic voltages. Simulation results illustrate good performance of the
single phase matrix converter based drive system. Therefore, this converter is
a good alternative to DC link converters at low output frequencies for
industrial applications. The converter has also an advantage of bi-directional
power flow between the input and load over diode rectifier front end
inverters.
The experimental set-up is based upon square wave PWM
technique, which is used for reducing harmonics and ripples in the output.
This technique, as compared to the sinusoidal PWM has reduced harmonic
content and hence less heat dissipation. As the output frequency increases,
lower order harmonics appear at the output. The results also show that the
single phase matrix converter provides quality output waveforms.
The simulated results of AC-DC-AC Converter, Indirect Matrix
Converter and Matrix Converter (Nine Switch Topology) based on several
modulation algorithms are compared and the results are validated. The power
quality studies based on THD levels are presented for different loading
conditions.
The hardware module of three phase matrix converter is realized by
employing Xilinx Spartan-3E XC3S100E FPGA controller for generating
SPWM signals to the IGBT switches. By keeping V/f ratio constant the
magnitude and frequency of the output voltage are controlled. The
experimental results of the matrix converter connected to R-Load and R-L
load are presented for different operating conditions. These matrix converters
can be employed in stand-alone wind energy conversion systems where the
output voltage or frequency control is required.
